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We classify nef vector bundles on a smooth hyperquadric of dimension three with first Chern class two over an algebraically closed field of characteristic zero. In particular, we see that they are globally generated.

We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ and give an explicit construction of these moduli as stratified spaces. We give sharp bounds for the…

Given a perverse sheaf on the moduli stack of principally polarized abelian varieties or the moduli stack of smooth curves with n marked points over a field of characteristic zero, we prove that the (orbifold) Euler characteristic is…

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the…

Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,\chi)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $\chi$. In…

For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the…

We study constraints on the Chern classes of a vector bundle on a singular variety. We use this constraint to study a variety which carries a Hodge cycle that are not a linear combination of Chern classes of vector bundles on it.

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles…

A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a…

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to…

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern…

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six…

We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values in an arithmetic Chow ring constructed by means…

We construct Chern-Simons bundles as $\mathrm{Aut}^{+}P$-equivariant $U(1)$ -bundles with connection over the space of connections $\mathcal{A}_{P}$ on a principal $G$-bundle $P\rightarrow M$. We show that the Chern-Simons bundles are…

Let X be a Hermitian locally symmetric space. We prove that every Chern class of X has a canonical lift to the cohomology of the Baily- Borel-Satake compactification X* of X and that the resulting Chern numbers satisfy the Hirzebruch…

Given a parabolic vector bundle, we construct for it a projectivization and tautological line bundle. These are analogs of the projectivization and tautological line bundle for an usual vector bundle. Using these we give a construction of…

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not…

We define a Chern--Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in $3$-forms modulo closed forms with integral cohomology class. We show an additivity property of this invariant for…

Boij-S\"oderberg theory concerns resolutions of graded modules over a polynomial ring over a field. Specifically Boij-S\"oderberg theory gives a description of the cone of Betti diagrams for Cohen-Macaulay modules. Eisenbud and Schreyer…

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's…